Fibonacci Phi - The Golden Ratio

[Fractal Freak]

Anyone else fascinated with the Golden Ratio (Phi)? It seems that there is an underlying principle with everything that is in this world that has some sort of aspect related to Phi. From artwork, proportions of the human body, to the growth rate of biological cells. Everything seems to have this infinite spiraling activity. Just curious if anyone else is as interested in it as I am.

 

[Zurtex]

A lot of people really like it, my favourite constant is by far e though.

 

[robert Ihnot]

I found it interesting that the phi is related to the 36-72-72 degree triangle. And also, the pentagram: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#363672
Once we can get the EDITED: cos(36) = $$\frac{1+\sqrt{5}}{4}$$, which is of quadratic form, we can find the sin(36) and then find the values for 6 degrees, and then using the half-angle formula find 3 degrees.
In general, that is as far as we can go in term of surds and whole degrees.
Gauss, of course, found the 17-gon sides in terms of surds, (http://www.absoluteastronomy.com/encyclopedia/h/he/heptadecagon.htm ) but that is very difficult to write out (doesn't evenly go into 360) and in one of his masterpieces, Disquisitiones Arithmeticaes published in 1801, he never went so far as to actually reach the final step in resolving it.

 

[Tide]

Once we can get the cos(36) = $$\frac{1+\sqrt{5}}{2}$$ ...

That can't be true ... unless someone added an imaginary part to 36 when no one was looking! ;)

 

[Tzar]

A lot of people really like it, my favourite constant is by far e though.

Why e? I think pi has to win hands down!

But phi is pretty cool too. Whats the weirdest place that anyone knows where phi comes up in?

 

[robert Ihnot]

Tide: Quote:
Originally Posted by robert Ihnot
Once we can get the cos(36) = ...
That can't be true ... unless someone added an imaginary part to 36 when no one was looking!

Sorry! It was divided by 4: $$cos(36)=\frac{1+\sqrt5}{4}$$ (I made the change in the original entry too.)

 

[Zurtex]

Why e? I think pi has to win hands down!
But phi is pretty cool too. Whats the weirdest place that anyone knows where phi comes up in?

I just think e is a lot prettier than pi, I also love all the different places it comes up in in basic calculus.

 

[Fractal Freak]

OK...I tried to go to the link that I posted and I don't know where the hell that came from...anyway, this is the actual link.

 

[Fractal Freak]

By the way, I think one of the most interesting places that phi is found is in the structure of DNA. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.
DNA in the cell appears as a double-stranded helix referred to as B-DNA.
This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms. (Cited from the Phi website)

The Fibonacci series is as follows: 0,1,1,2,3,5,8,13,21,34, etc.. which corresponds to Phi.

 

[tomfitzyuk]

A lot of people really like it, my favourite constant is by far e though.

Yeah e is much more fascinating, it appears in all sort of places. Most amazingly is places such as expotential growth and when you differentiate logorithmic functions. I suppose those two are similar but it's still amazing.

sin72/sin36 = Phi
2sin54 = Phi

These are things I found out myself :O

 

[Gokul43201]

OK...I tried to go to the link that I posted and I don't know where the hell that came from...anyway, this is the actual link.

That's a crackpot religious website. Links to such sites at not allowed in these Forums.

moderator's note: link removed

 

[robert Ihnot]

In case one has not gone through with it: 1,1,2,3,5,8,13,21... We look at the formula: $$S_n+S_(n+1)=S_(n+2)$$

This yields: $$\frac{S_n}{S_(n+1)}+1=\frac{S_(n+2)}{S_(n+1)}$$

Or: 1/X+1 =X implies $$X=\frac{1\pm\sqrt5}{2}$$
Taking the positive value,1.618...this ratio is so good that it easily finds the next Fibonacci number with round off, even starting with the second 1: 1xPhi=1.62=2, 2xPhi =3.236=3, 3xPhi=4.85=5, etc. And 21xPhi = 33.9=34, the next Fibonacci number.

We also have Binet's Formula for the Nth Fibonacci number:

$$F(N) =\frac{Phi^N (-1)^{N+1} Phi^{-N}}{\sqrt5}$$

 

[Tide]

You left out a + sign after the $\phi^n$ term.

 

[robert Ihnot]

Tide: You left out a + sign...

It seems clearer this way, thank you!

$$F(N) =\frac{Phi^N+ (-1)^{N+1} Phi^{-N}}{\sqrt5}$$

 

[Moo Of Doom]

It would probably look even better if you used \phi:
$$F(N) =\frac{\phi^N+ (-1)^{N+1} \phi^{-N}}{\sqrt5}$$
:)

 

[robert Ihnot]

Moo of Doom: It would probably look even better if you used \phi:

That's a good idea. Live and learn I always say!


$$F(N) =\frac{\phi^N+ (-1)^{N+1} \phi^{-N}}{\sqrt5}$$

 

[Tide]

I think it looks better this way:

$$F_n = \frac {\phi^n - (-\phi)^{-n}}{\sqrt 5}$$

;)

 

[robert Ihnot]

Tide: I think it looks better this way:

Certainly does. Actually my form came about late at night when I was afraid that a lot of latex would be written if I had to invert Phi and rationalize the denominator, etc. So I looked for a way out.

In case of further difficulity over the question of capitalizing Phi, I will quote:
"We will call the Golden Ratio (or Golden number) after a greek letter,Phi () here, although some writers and mathematicians use another Greek letter, tau (). Also, we shall use phi (note the lower case p) for a closely related value." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#golden

 

[matt grime]

Whilst phi certainly does have things to do with nature, whcih are well known and understood (there is a notion of speed of convergence and phi has some special properties there that mean sunflower seeds for example are dstributed approximately as a fibonnacci sequence; this ensures best exposrure for least coverage) but most of what you cite as frequent occurences are famously red herrings. There is no use of phi in art or sculpture by the greeks: pots came in many sizes, and there is no reason to suppose that rations of phi have ever been thought "more attractive". Anatomical references are not accurate either. It so happens that if you divide something into two roughly equal parts then they may lbe close to phi, an number that is close to 1.

 

[Mike AkA Don]

I too find it fascinating. http://goldennumber.net/ has quite a lot on it and relates it to art it really is incredible. I did quite a lot of research on it after reading The Da Vinci Code.